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    Wyświetlanie 1-9 z 9
    Tytuł:
    On The Co-Roman Domination in Graphs
    Autorzy:
    Shao, Zehui
    Sheikholeslami, Seyed Mahmoud
    Soroudi, Marzieh
    Volkmann, Lutz
    Liu, Xinmiao
    Powiązania:
    https://bibliotekanauki.pl/articles/31343438.pdf
    Data publikacji:
    2019-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    co-Roman dominating function
    co-Roman domination number
    Roman domination
    Opis:
    Let $G = (V, E)$ be a graph and let $f : V (G) \rightarrow {0, 1, 2}$ be a function. A vertex $v$ is said to be protected with respect to $f$, if $f(v) > 0$ or $f(v) = 0$ and $v$ is adjacent to a vertex of positive weight. The function $f$ is a co-Roman dominating function if (i) every vertex in $V$ is protected, and (ii) each $ v \in V $ with positive weight has a neighbor $ u \in V $ with $ f(u) = 0 $ such that the function $ f_{uv} : V \rightarrow {0, 1, 2} $, defined by $ f_{uv} (u) = 1$, $ f_{uv}(v) = f(v) − 1$ and $ f_{uv}(x) = f(x)$ for $ x \in V \backslash \{ v, u \} $, has no unprotected vertex. The weight of $f$ is $ \omega(f) = \Sigma_{ v \in V } f(v) $. The co-Roman domination number of a graph $G$, denoted by $ \gamma_{cr}(G) $, is the minimum weight of a co-Roman dominating function on $G$. In this paper, we give a characterization of graphs of order $n$ for which co-Roman domination number is \( \tfrac{2n}{3} \) or $n − 2$, which settles two open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1–10]. Furthermore, we present some sharp bounds on the co-Roman domination number.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 2; 455-472
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Roman bondage in graphs
    Autorzy:
    Rad, Nader
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/743601.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    domination
    Roman domination
    Roman bondage number
    Opis:
    A Roman dominating function on a graph G is a function f:V(G) → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f(V(G)) = ∑_{u ∈ V(G)}f(u)$. The Roman domination number, $γ_R(G)$, of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_R(G)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E' ⊆ E(G) for which $γ_R(G -E') > γ_R(G)$. We determine the Roman bondage number in several classes of graphs and give some sharp bounds.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 4; 763-773
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Double Roman Domatic Number of a Digraph
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31348166.pdf
    Data publikacji:
    2020-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    digraph
    double Roman domination
    double Roman domatic number
    Opis:
    A double Roman dominating function on a digraph $D$ with vertex set $V(D)$ is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function $f : V (D) → {0, 1, 2, 3}$ having the property that if $f(v) = 0$, then the vertex $v$ must have at least two in-neighbors assigned 2 under $f$ or one in-neighbor w with $f(w) = 3$, and $if f(v) = 1$, then the vertex v must have at least one in-neighbor $u$ with $f(u) ≥ 2$. A set ${f_1, f_2, . . ., f_d}$ of distinct double Roman dominating functions on $D$ with the property that $∑_{i=1}^df_i(v)≤3$ for each $v ∈ V (D)$ is called a double Roman dominating family (of functions) on $D$. The maximum number of functions in a double Roman dominating family on $D$ is the double Roman domatic number of $D$, denoted by $d_{dR}(D)$. We initiate the study of the double Roman domatic number, and we present different sharp bounds on $d_{dR}(D)$. In addition, we determine the double Roman domatic number of some classes of digraphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 4; 995-1004
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Roman Domatic Problem in Graphs and Digraphs: A Survey
    Autorzy:
    Chellali, Mustapha
    Rad, Nader Jafari
    Sheikholeslami, Seyed Mahmoud
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/32304148.pdf
    Data publikacji:
    2022-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Roman domination
    domatic
    Opis:
    In this paper, we survey results on the Roman domatic number and its variants in both graphs and digraphs. This fifth survey completes our works on Roman domination and its variations published in two book chapters and two other surveys.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 3; 861-891
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Weak signed Roman k-domination in digraphs
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/29519480.pdf
    Data publikacji:
    2024
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    digraph
    weak signed Roman k-dominating function
    weak signed Roman k-domination number
    signed Roman k-dominating function
    signed Roman k-domination number
    Opis:
    Let $ k ≥ 1 $ be an integer, and let $ D $ be a finite and simple digraph with vertex set $ V (D) $. A weak signed Roman k-dominating function (WSRkDF) on a digraph $ D $ is a function $ f : V (D) → {−1, 1, 2} $ satisfying the condition that $ \Sigma_{x∈N^−[v]} f(x) ≥ k $ for each v ∈ V (D), where $ N^− [v] $ consists of $ v $ and all vertices of $ D $ from which arcs go into $ v $. The weight of a WSRkDF $ f $ is $ w(f) = \Sigma_{v∈V} (D) f(v) $. The weak signed Roman k-domination number $ \gamma_{wsR}^k (D) $ is the minimum weight of a WSRkDF on $ D $. In this paper we initiate the study of the weak signed Roman k-domination number of digraphs, and we present different bounds on $ \gamma_{wsR}^k (D) $. In addition, we determine the weak signed Roman k-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number $ \gamma_{wsR} (D) = \gamma_{wsR}^1 (D) $ and the signed Roman k-domination number $ \gamma_{sR}^k (D) $.
    Źródło:
    Opuscula Mathematica; 2024, 44, 2; 285-296
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Signed Total Roman Domination in Digraphs
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31342127.pdf
    Data publikacji:
    2017-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    digraph
    signed total Roman dominating function
    signed total Roman domination number
    Opis:
    Let $D$ be a finite and simple digraph with vertex set $V (D)$. A signed total Roman dominating function (STRDF) on a digraph $D$ is a function $ f : V (D) \rightarrow {−1, 1, 2} $ satisfying the conditions that (i) $ \Sigma_{x \in N^− (v) } f(x) \ge 1 $ for each $ v \in V (D) $, where $ N^− (v) $ consists of all vertices of $D$ from which arcs go into $v$, and (ii) every vertex u for which $f(u) = −1$ has an inner neighbor $v$ for which $f(v) = 2$. The weight of an STRDF $f$ is $ w(f) = \Sigma_{ v \in V } (D) f(v) $. The signed total Roman domination number $ \gamma_{stR} (D) $ of $D$ is the minimum weight of an STRDF on $D$. In this paper we initiate the study of the signed total Roman domination number of digraphs, and we present different bounds on $ \gamma_{stR} (D) $. In addition, we determine the signed total Roman domination number of some classes of digraphs. Some of our results are extensions of known properties of the signed total Roman domination number $ \gamma_{stR} (G)$ of graphs $G$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 1; 261-272
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Signed Total Roman k-Domatic Number Of A Graph
    Autorzy:
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31341581.pdf
    Data publikacji:
    2017-11-27
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    signed total Roman k-dominating function
    signed total Roman k-domination number
    signed total Roman k-domatic number
    Opis:
    Let $ k \ge 1 $ be an integer. A signed total Roman $k$-dominating function on a graph $G$ is a function $ f : V (G) \rightarrow {−1, 1, 2} $ such that $ \Sigma_{ u \in N(v) } f(u) \ge k $ for every $ v \in V (G) $, where $ N(v) $ is the neighborhood of $ v $, and every vertex $ u \in V (G) $ for which $ f(u) = −1 $ is adjacent to at least one vertex w for which $ f(w) = 2 $. A set $ { f_1, f_2, . . ., f_d} $ of distinct signed total Roman $k$-dominating functions on $G$ with the property that $ \Sigma_{i=1}^d f_i(v) \le k $ for each $ v \in V (G) $, is called a signed total Roman $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed total Roman $k$-dominating family on $G$ is the signed total Roman $k$-domatic number of $G$, denoted by $ d_{stR}^k (G) $. In this paper we initiate the study of signed total Roman $k$-domatic numbers in graphs, and we present sharp bounds for $ d_{stR}^k (G) $. In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman $k$-domatic number of some graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 4; 1027-1038
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Signed Roman Edge k -Domination in Graphs
    Autorzy:
    Asgharsharghi, Leila
    Sheikholeslami, Seyed Mahmoud
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31342188.pdf
    Data publikacji:
    2017-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    signed Roman edge k -dominating function
    signed Roman edge k -domination number
    Opis:
    Let $ k \ge 1 $ be an integer, and $ G = (V, E) $ be a finite and simple graph. The closed neighborhood $ N_G [e]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and all edges having a common end-vertex with $e$. A signed Roman edge $k$-dominating function (SREkDF) on a graph $G$ is a function $ f : E \rightarrow {−1, 1, 2} $ satisfying the conditions that (i) for every edge $e$ of $G$, $ \Sigma_{ x \in N_G [e] } f(x) \ge k $ and (ii) every edge e for which $f(e) = −1$ is adjacent to at least one edge $ e^′ $ for which $ f(e^′) = 2 $. The minimum of the values $ \Sigma_{e \in E} f(e) $, taken over all signed Roman edge $k$-dominating functions $f$ of $G$ is called the signed Roman edge $k$-domination number of $G$, and is denoted by $ \gamma_{sRk}^' (G) $. In this paper we initiate the study of the signed Roman edge $k$-domination in graphs and present some (sharp) bounds for this parameter.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 1; 39-53
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Bounds on the Signed Roman k-Domination Number of a Digraph
    Autorzy:
    Chen, Xiaodan
    Hao, Guoliang
    Volkmann, Lutz
    Powiązania:
    https://bibliotekanauki.pl/articles/31343713.pdf
    Data publikacji:
    2019-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    signed Roman k-dominating function
    signed Roman k-domination number
    digraph
    oriented tree
    Opis:
    Let $k$ be a positive integer. A signed Roman $k$-dominating function (SRkDF) on a digraph $D$ is a function $ f : V (D) \rightarrow \{−1, 1, 2 \} $ satisfying the conditions that (i) $ \Sigma_{ x \in N^− [v] } f(x) \ge k $ for each $ v \in V (D) $, where $ N^− [v] $ is the closed in-neighborhood of $v$, and (ii) each vertex $u$ for which $f(u) = −1$ has an in-neighbor $v$ for which $f(v) = 2$. The weight of an SRkDF $f$ is $ \Sigma_{ v \in V (D) } f(v) $. The signed Roman $k$-domination number $ \gamma_{sR}^k (D) $ of a digraph $D$ is the minimum weight of an SRkDF on $D$. We determine the exact values of the signed Roman $k$-domination number of some special classes of digraphs and establish some bounds on the signed Roman $k$-domination number of general digraphs. In particular, for an oriented tree $T$ of order $n$, we show that $ \gamma_{sR}^2 (T) \ge (n + 3)//2 $, and we characterize the oriented trees achieving this lower bound.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 1; 67-79
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-9 z 9

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